A general scaling law can be derived for the relative momentum deflection produced on a particle beam by fringe fields, to leading order. The formalism is applied to two concrete examples, for magnets having dipole and quadrupole symmetry. During recent years, the impact of magnet fringe fields is becoming increasingly important for rings of relatively small circumference but large acceptance. A few years ago, following some heuristic arguments, a scaling law was proposed (1), for the relative deflection of particles passing through a magnet fringe-field. In fact, after appropriate expansion of the magnetic fields in Cartesian coordinates, which generalizes the expansions of Steffen (2), one can show that this scaling law is true for any multipole magnet, at leading order in the transverse coefficients (3). This paper intends to provide the scaling law to estimate the impact of fringe fields in the special cases of magnets with dipole and quadrupole symmetry.
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